Let G be a graph with vertex set V = { v1, v2, v3, v4, v5}.
Is it possible for the degrees of the vertices to be 3, 6, 2, 1, 5, respectively? Why or why not?
See your next post to find out how to calculate the number of edges.
Hence deduce if such a graph can exist or not.
MathMate: Is this correct for the following problem?
3 + 6 + 2 + 1 + 5
2E = 17
E = 8.5
If your allowed to have a decimal as an answer, then to come up with the answer for the amount of edges then yes it is possible for the degrees of the vertices to be 3, 6, 2, 1, 5. If your not allowed to have a decimal as an answer for the amount of edges, then no its not possible for the degrees of the vertices to be 3, 6, 2, 1, 5.
To determine if it is possible for the degrees of the vertices in a graph to be a specific sequence, we can use the Handshaking Lemma.
The Handshaking Lemma states that in any undirected graph, the sum of the degrees of all the vertices is equal to twice the number of edges in the graph.
In this case, the degrees of the vertices are given as 3, 6, 2, 1, 5. Let's calculate the sum of these degrees:
3 + 6 + 2 + 1 + 5 = 17
According to the Handshaking Lemma, the sum of the degrees should be twice the number of edges. Let's assume the number of edges in the graph is 'E'.
Therefore, 17 = 2E
Simplifying the equation, we get:
E = 8.5
The number of edges, 'E', has to be a whole number since fractional edges are not possible. However, the value of E we obtained is 8.5, which is not a whole number.
Therefore, it is not possible for the degrees of the vertices to be 3, 6, 2, 1, 5, respectively, in a graph with vertex set V = {v1, v2, v3, v4, v5}.